Review: "Math Trailblazers"
The following review of the Kendall/Hunt "Math Trailblazers" program was written by a parent in Colorado.
A couple of notes about the abbreviations used:
In the original of this article, the author referred to this program as "KHM" -- it is changed here to "MTB" to match the way the vendor refers to it. Also, the author frequently offers comparisons to Houghton-Mifflin Mathematics, which she refers to here as "HMM".
Comments? Write to us at firstname.lastname@example.org. We'd especially welcome additional reviews or reports on this math program.
A review of
IN Math Trailblazers
OUT of Math Trailblazers
All strategies are very, very equal
All strategies are not very equal
Start with "the tens"
Start with "the ones"
Meet children where they are at
Take children where they can go
Do Less, Learn More
Do More, Learn More
Science Experiments with Marshmallows
Vertically Integrated Mathematics
Class Moves On before kids get bored
A Child Stays Put until he understands it
A lot of Group Work
A lot of Individual Work
Effortless Arithmetic Computation on Paper
Feeling Mathematically Powerful
Mathematically Correct Sentences
A page in a journal
A page of problems
Parents that actively listen
Parents that actively teach
When you read the advertising on the front of the package, everything IN Math Trailblazers sounds pretty wonderful. But you have to read the ingredient list on the side of the package to find out what Kendall/Hunt left OUT of Math Trailblazers.
Parents are ultimately responsible for the quality of their childrens education. Ten years from now a parent shouldnt say "But the teacher promised this series would lay a solid foundation for my childs conceptual understanding of mathematics ."
My twin boys will start First Grade this fall at a public elementary school in Colorado. My boys are bright, but not exceptional. Their education will be their full-time job for the next twelve years. And they will have to bring work home most nights, so I believe I owe it to them to try to participate in choosing the best possible mathematics curriculum for them. Our school district does not make it easy to participate in curriculum selection. A district "expert panel" is convened which is composed of teachers who are parents, but which does not include parents who are not teachers. Earlier this year, the expert panel suggested a short list of suitable curricula from which each school could choose their favorite program. Parents were allowed to review the schools choice, but our reviews have had no defined value in the process. There wasnt a groundswell of interest. Unfortunately, many parents thought that they werent qualified to consider math textbooks. I keep telling them, "if you graduated from Second Grade, youre qualified."
I spent one weekend studying Unit 11, "Ways of Subtracting Larger Numbers" from Kendall/Hunts Math Trailblazers (MTB) series for the Second Grade. I had the Teachers Implementation Guide (TIG), the Student Activity Book, and the Unit Resource Guide (URG) in front of me. There is no textbook for children, just simple word problems in an activity book. Thus I had to read the suggested teacher scripts from the Unit 11 URG if I wanted to find out how MTB is taught. As Kendall/Hunt will tell you in their promotional literature, this math series has been years in the making, and has been extensively field-tested at hundreds of schools. I am sure that they are sure that the teacher script in the URG is the ideal script.
My goal was to hear and see for myself the difference between how a whole number operation, such as Subtraction, is taught in the MTB series, compared to how it is taught in the Houghton-Mifflin Mathematics (HMM) series. I was concerned because our school staff proposes to split the math curriculum between MTB in grades K-2 and HMM in grades 3-5. I had no idea why they would do that. It sounded like they were going to cut the baby, my childrens education, in half.
Some would call HMM a more traditional approach to teaching arithmetic. Being an old math major, I thought it looked sufficiently rigorous for my boys, who are going to spend all day at school, five days a week. HMM has all the bells and whistles, all the games and manipulatives, that are supposed to make it fun for kids today. What was wrong with it?
You might think I was kidding if I told you that what was wrong with HMM is that it teaches children to "start with the ones" when they add or subtract two-digit numbers. What educators think is right with MTB is that it teaches children to "start with the tens."
Educators believe that starting with the tens is a natural way for Second Grade children to compute with two-digit numbers, and therefore it is good. Specifically, it is good enough. At least it is better than what happens to children when teachers teach the standard written arithmetic procedures by rote.
I think there is some high ground in the middle.
Teachers could tell students about starting with the tens and about starting with the ones. The former is our preferred approach when were standing in front of the shelves at the grocery store and the dollar amounts are low. The latter is our preferred approach when we sit down with a pencil and paper and numbers of any size. To save time, we start with the ones. But our children also need to know why we can start with the ones.
Teachers should teach standard written arithmetic procedures in the primary grades, and our teachers should try to do a very good job, because:
It is necessary to write out, in detail, the teacher script for the lessons in Unit 11 in order to explain what is wrong with MTB. Im sorry. It does no good to look at the student activity book. My summary of the text from MTB is in italics, and quoted where exact.
Lesson 1: "Subtraction Seminar," 2 days, TIG suggests one hour per day
The teacher writes several two-digit subtraction problems on the board. The class is to share ideas as to which ones are easy or hard. "Explain that it is acceptable to have different opinions." There is a journaling opportunity.
This seems like a great opportunity for children to put on their thinking caps, and for the teacher to gauge where the children are at with respect to subtraction of larger numbers. This Unit is laid out like a previous unit on two-digit addition.
Will the children think the problems with the larger numbers are harder? Will they recognize that they can use simple strategies such as counting if the numbers are large but close, or if the number to be subtracted is very small?
The teacher is not supposed to say which is easy or hard. Looking ahead to the end of this unit, will the children have an opportunity to reflect on which problems really were easy, and which really were hard?
The problem for the whole class to discuss is: what happens if you use $1 to buy a 39-cent item? What change should you get back? The teacher can expect strategies such as
These counting strategies seem richly varied. They exhibit a wide developmental range.
The fourth strategy is to count up from 1 to 100 with the aid of counters, then to count up from 1 to 39 and remove 39 counters, and then to count up from 1 to 61.
The third strategy is to count down by 39 with the aid of a chart: 100 39 = 100, 90, 80, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61. The teacher could point out that this child made use of the fact that 39 decomposes into 3 tens and 9 ones, and the strategy could be written:
100 39 = 100 30 9 = 100 10 10 10 1 1 1 1 1 1 1 1 1 = 61.
The second strategy is to count up by tens then ones with the aid of fingers. This child understands that a b = c is the same as c + b = a, and then makes use of the fact that 61 decomposes into 6 tens and 1 one. It could be written:
39 + 10 + 10 + 10 + 10 + 10 + 10 + 1 = 100.
There is no guidance in the URG or the TIG as to whether the teacher should stop now and lead a short discussion about the algebraic concept behind her strategy.
The first strategy is not just a counting strategy, and it is possibly the most advanced in the sense that the child stops to trade one ten for ten ones before counting further. Given the time, the teacher could see if that child could express her strategy in simple mathematical sentences. The teacher could write out what the child had assumed so that the rest of the class could follow the reasoning:
1 dollar = 10 dimes (and 39 cents = 3 dimes + 9 pennies.)
10 dimes 3 dimes (= 7 dimes.)
(7 dimes = 6 dimes + 1 dime and) 1 dime = 10 pennies.
(6 dimes + 10 pennies 9 pennies =) 6 dimes + 1 penny = 61 cents.
That is a lot to write, and each step is not properly justified. I have no idea if this child is constructing her own mathematical understanding right there in the classroom, or if her mother has been working on subtraction problems with her at home, or if her grandma had a chat with her about money and making change, and one of them spilled the beans about trading one ten for ten ones.
In the hour that is allotted for math each day, I have no sense that the teacher could do more than acknowledge each of 20 different strategies, and reassure each child that if they got the number "61," they have a valid strategy.
It appears that from both the URG and the TIG it is not okay for the teacher to single out the first or second strategies as any better or more interesting and worthy of discussion than the others. There is no direction from the URG that the teacher might have children write their strategies in complete mathematical sentences.
Homework: At the Playground. N children are playing at the playground, if X are boys, how many are girls? If Y are swinging, how many are not swinging? Etcetera.
The letter home to the parents says "It is important that your child understands that the traditional approach for solving problems is just one of many acceptable methods of finding the answer. Other problem-solving strategies your child will explore include estimating, drawing pictures, and using tools such as base-ten pieces and calculators."
That sounds like a gentle warning to parents: Dont spill the beans about writing down the problem columnwise on paper, decomposing the numbers, and regrouping when necessary. That must be what they mean by the "traditional approach". The message is that your child needs to discover subtraction for herself. You shouldnt say qualitative things such as how youve always started with the ones and "borrowed from the tens" since you were in Second Grade. You would ruin it for your child.
Lesson 2: "Is It Reasonable?" 1 hour
The problem for the whole class to discuss is: I used $1 to buy a 29-cent pretzel, and I got 2 quarters and 1 nickel back. My change should have been about 70 cents. How do I know they gave me the wrong change?
The teacher can expect strategies (that I have simplified) such as
Class work consists of five problems.
My favorite answer is the fourth answer, although the strategy is not robust.
I was amazed that MTB put estimation into the unit right here. It took me a very long time to wrap my mind around it. The simple answer is that this MTB unit on two-digit subtraction is laid out like the unit on two-digit addition.
The more complex answer is that MTB is trying to get children to consider "the tens" before "the ones." The standard written procedure for subtraction that starts with "the ones" has not been presented to the children. This estimation exercise forces children to consider "the tens" first. The teacher is reminded to discourage children from solving the equations exactly. Children are told not to worry about "the ones."
I looked at the class work, and none of the problems gets the children into deep water, forcing them to decide what, say, 24, 25, or 26 are near to.
A quick look at the HMM text shows the lesson on estimation comes after lessons on tens subtraction (why 100 30 is like 10 3) and after exact subtraction, where the children do worry about "the ones." That seems more logical to me. I thought estimation is what you do when you are trying to change an exact tens and ones problem to its nearest tens subtraction problem, which you then solve with single digits.
To me, its a sophisticated thing to simplify a problem appropriately, to know what numbers in the ones column you can safely ignore and still get a good estimate. It requires discretion. Which is why neither MTB nor HMM have the children estimate something like 56 24. A child should not leave this unit thinking they can estimate all two-digit subtraction problems.
Lesson 3: "Base-ten Subtraction," 2 hours
The teacher has the children work in pairs and use manipulatives (a base-ten board) to solve the problem 78 42. She is to ask, "Will the answer be more or less than 70? Estimate a reasonable answer."
"Give students one or two other problems that involve regrouping. Before subtracting, they should again make an estimate of the difference. Ask the students to share how they might have solved the subtraction problem without base-ten pieces. (It remains important to continue to validate other methods.)"
Class work consists of 4 problems. First they must estimate. Then they must do it "with base-ten pieces" and "one other way."
Using proportionate manipulatives is a compelling way to visually demonstrate number composition, decomposition, and regrouping from the tens.
The teacher directs children to estimate, which means she is directing them to think about the tens first, and therefore the reasonableness of their final answer.
In the class work, Ill assume the "one other way" must be a counting strategy, like those sample strategies given in Lesson 1, and that each child will write it down somehow, using some English.
Second graders who still have a lot to learn about writing English are forced to use a lot of English words to describe what they are doing as they work with manipulatives and invent counting strategies, rather than being encouraged to represent their work with numbers in proper mathematical sentences.
A quick look forward and back through the Student Activity Book comes up with very few pages of problems. Presumably MTB reduces the quantity of problems to focus children on the quality of their problem solving. However, if MTB thought to rid itself of tedious pages of problems, they are substituting a lot of tedious writing. Apparently, much of the time for Lesson 3 is spent on English literacy. I think more time should be spent on Mathematical numeracy. Teachers who are trying to read and respond constructively to the mathematics represented in the childrens activity books and journals may have little time to comment on the quality of the English. It might be counterproductive to have our children write more English under these circumstances.
Consider an exceptional case. I recall my brothers experience in school. He is dyslexic; he wasnt diagnosed until the Fifth Grade. He can read "in" the words and numbers, his difficulty is in writing "out" the words and numbers. Obviously, in todays schools, he would be diagnosed much sooner. I can imagine his IEP would include a note from the learning specialist, "Please encourage Lawrence to write in short mathematical sentences. 2 + 2 is a more reasonable goal for Lawrence than I thought first, I had two, then I add two more."
Lesson 4: "Paper-and-Pencil Subtraction," 2 hours
As a warm-up, the teacher puts the problem 35 + 36 on the overhead, and asks for ideas to solve it. "Emphasize the idea that no particular problem-solving strategy is better than another and that students should use whatever method they find most helpful."
Then, the teacher puts the problem 60 25 on the overhead, and she asks for suggestions on how to solve it. If nobody suggests base-ten pieces, the teacher suggests base-ten pieces and demonstrates regrouping.
Next, the teacher has student pairs solve the problem 42 18. The children review their answers as a group.
Finally, the teacher puts the problem 53 29 = 24 on the overhead, in vertical form. The compact notation to describe the regrouping in our standard, right to left subtraction procedure, is written: 5 crossed out and a small 4 above, and 3 crossed out with a small 13 above. The teacher asks the children what they think happened. Then the teacher demonstrates what happened, using base-ten pieces.
Class work consists of 8 problems, 4 of which are word problems. There is no requirement to use the compact paper-and-pencil description of our standard procedure. The children need to "draw a picture or write to explain your thinking."
A suggested game is "Difference Wars" where the children play in pairs. Each child is dealt a hand of four single-digit number cards, and the child who constructs the largest two-digit subtraction difference wins the hand. The teacher is told to "Observe their methods and correct any errors you notice." The children are then to write in their journals about their experience playing this game.
The assessment page for this lesson is "Many Ways to find the Answer". The children are to find 3 ways to solve one subtraction problem.
I dont know whether to laugh or to cry. I am amazed that the URG suggests that this lesson will take 2 hours. I would expect many, many more hours to do it justice.
The brief appearance of our standard written subtraction procedure is apparently made only to satisfy the letter, but not the spirit, of our state laws that set forth standards and benchmarks for Second Grade mathematics content that include teaching children how to start with "the ones."
Let me make sure we are all understand the semantics. For the purposes of adults discussing how children do arithmetic, a "procedure" is a "strategy" is an "algorithm." A procedure is a sequence of steps to solve an arithmetic problem. "Strategy" sounds a whole lot better. It sounds very creative. Parents love to hear that their child is strategizing to solve math problems in the classroom. A more accurate term would be "tactic." "Algorithm" sounds a whole lot worse than "strategy," doesnt it? "Algorithm" sounds very robotic. It sounds like something a computer does, not what a human child should be doing in the classroom. But "algorithm" is the term mathematicians use to describe a sequence of steps. In this review I will keep writing about "procedures," in an effort to stay neutral. Our standard written subtraction procedure belongs to all of us. I dont think of it as a cultural artifact. It is an intellectual treasure held in common.
The title of this lesson, "Paper-and-Pencil Subtraction", is barely civil. It is pejorative. Perhaps MTB was trying to make paper and pencil sound less scary to children. But at the start of each days literacy block, does the teacher say, "Okay, kids, lets do some paper-and-pencil English?"
I would prefer the teacher said, "Lets write down, in mathematical sentences, what were doing with the base-ten pieces." There are rules for writing mathematics. "53 29 = 24" is written very well. I want my boys to see and hear lots of mathematics, and I want them to learn to read and write in mathematical sentences. I dont care if they are considering the tens first, or the ones. I want them to write numbers. They will then be numerate.
Suppose the children at our school had a Greetings Seminar in their Spanish class. If the teacher asked a child, "Como estas?" should he answer "Muy bien" or "Im fine" or even "Muy okay?" Likewise, our mathematics curriculum should encourage proper spoken and written mathematical responses from our children. It seems to me that sloppy speech will only make learning math that much harder.
Regarding the class work in Lesson 4, notice that children are not required to try out our standard procedure with paper and pencil, starting with the ones. They can use any of their previous counting strategies. They can count up by ones. They can count down by ones. They can count up by three and down by two. They can start with the tens or start with the ones. They can use hash marks and they can draw pictures.
The URG urges the teacher to emphasize to children that no strategy is better than another. This urgent message is included in every lesson. But my personal experience tells me that some strategies are better than others because of their speed or reliability, or because they are a basis for further conceptual or procedural work. A childs use of a particular strategy tells the teacher exactly where the child is at on their mathematical journey from point A, the concrete, to point B, the abstract. Its not clear when, if ever, MTB has a teacher apply this information to help a child advance along the path.
I think MTB finds itself in a bind. The teacher assiduously collects the childrens strategies, and then just as assiduously she must avoid making quality distinctions among the strategies. Perhaps the children make quality distinctions. The URG does not suggest a class vote on the best strategy for solving the days subtraction problem.
Its really quite noble that the folks at Kendall/Hunt want our children to feel very equal to each other. "All men are created equal." One child is not better than another child. But their equality in the sight of God should not blind us to the fact that children arrive at the start of Second Grade with unequal mathematical experiences. Each child is at a different point on his or her mathematical journey, and our only compassionate action must be to help every child move further down the path to mathematical mastery. Isnt that mastery written in black and white in our state standards and benchmarks for Second Grade?
Do the good folks at Kendall/Hunt fear that if we order and compare the childrens strategies by talking about good, better, and best strategies, that we will also order and compare our children, and that will silence our children?
I fear that there are damaging consequences if we teach our children to believe that mathematics is subjective, that it is not outside of ourselves, that mathematics is an individual expression made within the confines of a peer group. Mathematics is expressed by individuals, but it is not a form of individual expression.
Of course we want our children to independently discover as much as they can about math, but we dont just throw them all into the pool and watch to see which children swim to the side at the end of the school year. We must act responsibly. We must give them small, manageable problems in an incrementally optimal order. Which reminds me of a specific episode in preschool art of developmentally appropriate practice: Let the children explore the process of using three different paint colors to make a relief print with bubble wrap; let the teacher choose the three different paint colors.
One of the consequences of spending so much time on so many different strategies is that some children are paralyzed by so many choices, as at a strategy buffet. Still other children put pieces and parts of strategies together, from what their peers suggest in the classroom, but the cobbled strategy does not work. There are children who learn more effectively if they are presented with fewer choices.
If I may suggest one beautiful thing about our standard subtraction procedure: once a child masters it, it is hers forever, and forever she will be on an equal footing with her classmates, with her mother, her brother, and her grandmother. It doesnt matter how big the numbers get.
Returning to the teacher script for Lesson 4, there is no suggestion that now might be a good time to revisit the strategies suggested by the children in Lesson 1. A teacher could point out connections between childrens strategies, and how they do or do not relate to our standard arithmetic procedures. Now would be a good time to go back and decide which of the problems at the beginning of the unit were "easy" and which were "hard."
Perhaps the main difficulty with the MTB approach is that a teacher is responsible for twenty or more children in her classroom, not just two or three. MTB may be best implemented in a small group setting, with one teacher and just a few creative children.
The Difference War game sounds fantastic. It sounds like the most powerful teaching tool in Unit 11. To be dealt the numbers 7, 8, 6, and 1, and to have to start thinking about the tens AND the ones I guess Id pick 87 16 = 71. Two things concern me about the games implementation, though. First, with a classroom of twenty children there will be ten pairs, and the teacher is told to "observe their methods and correct any errors you notice." I think a teacher would be well occupied to keep track of the work of one pair. Secondly, the children are supposed to privately journal about this activity. Why not facilitate a class discussion to verify who got the biggest difference? Who got the smallest difference? This game could easily occupy the whole hour, especially if the children play long enough to allow the teacher to observe each childs method.
Lesson 5: "Snack Shop Addition and Subtraction," 1 hour
A menu for a snack shop is provided. There is a warm-up problem. "Read the problem and solve it as a class. Have students explain their thinking and demonstrate how they would solve the problem. Some may solve it in their heads, while others choose to use counters, the 200 chart, base-ten pieces, or paper and pencil. Some students may suggest the use of the calculator. Distribute calculators and introduce keystrokes for subtraction. Tell the students to pretend they each have $2."
I hope they are not learning those keystrokes by rote! I hope the kids use their calculators with some understanding of the electronics!
I was excited when I started to read about the Snack Shop; I thought this would be the opportunity for individual children to discover if their individual counting strategy was efficient, by being presented with more than three or four problems at a time. Children could experience the logical consequences of their strategy choice. But the teacher hands out calculators. It is true that the purpose of this unit was to discover subtraction, to explore ways of subtracting, but not to practice subtracting or develop an efficient subtraction strategy.
Calculators certainly do level the playing field for all children. But instead of leveling the playing field by handing out calculators, why not help each child develop an efficient subtraction strategy, or maybe even master our standard written subtraction procedure? Thats an old-fashioned way to level the playing field.
Back in the MTB Teacher Implementation Guide, p. 182, I unearthed this description of how MTB teaches subtraction:
"Later in grade 2, systematic work begins on paper-and-pencil methods for subtracting two-digit numbers. Students are asked to solve two-digit subtraction problems using their own methods and to record their solution on paper. The class examines and discusses the various procedures that students devise. At this time, if no student introduces the standard subtraction algorithm, then the teacher does so, explaining that it is a subtraction method that many people use. The standard method is examined and discussed, just as the invented methods were. Students who do not have an effective method of their own are urged to adopt the standard method."
Take a look at how children learn about starting with "the ones" in HMM Chapter 6. These topics are covered in this order:
The Teachers Guide suggests a warm-up activity before each lesson, and each lesson plan has two different discovery activities to choose from. One activity is a cooperative kinesthetic/tactile activity for pairs, small groups, or the whole class, and the other activity is a class discussion of a subtraction problem. Thats where the games are.
Each child has a "consumable" workbook that is really a textbook. Each lesson of their textbook starts with a simple concept statement at the top of the page, written at a Second Grade level, and there is a pictorial or symbolic demonstration of the concept.
Through the use of "ten-frames," every subtraction problem can be set up column-wise with the tens and the ones named, to help a child remember place value during the move to this symbolic representation. If a child brings home the HMM textbook, the parent can know exactly what is going on in the classroom, and the parent can help teach it.
Each lesson ends with guided practice so that children learn to write correct mathematical sentences. There is no journaling. After practice, children can solve an interesting word problem. For example, Lesson 8 ends with "data" on birds that children use to create their own subtraction problem.
For more ideas on how to teach a child how to start with the ones, see Appendix A.
There are many things in Houghton-Mifflin Mathematics for a mother (or a father) to love. HMM is not unique in this sense, there are many other fine mathematics series available for K-6 that have the following features:
What is the point of radically de-emphasizing our standard written procedures? Where did this bright idea come from? Rousseau. See E.D. Hirschs article in Education Next, http://www.educationnext.org/2001sp/34.html. But closer to home, Constance Kamii has been pushing a constructivist ideology in primary mathematics education for many years. Many teachers graduating from schools of education today know only a little math, but a lot about Kamiis work with children. Kamii takes it as an article of her faith in Piaget that children can reinvent arithmetic, that standard written procedures are neither required nor desired in the primary classroom. See Appendix D.
MTB is paradoxical. It is an example of extreme differentiation. One child might know how all the mental arithmetic strategies that start with the tens work, but she still prefers to use our standard procedure with paper and pencil. Another child never feels comfortable with the mental arithmetic strategies proposed by his classmates, and instead he prefers to count by ones all the way through this unit. These children coexist in the same classroom, without quality distinctions. They could be chosen as a pair to play the Difference War game. Thats a differentiated classroom. Each childs preference is accommodated all the way to the bitter end.
Which means that MTB is completely undifferentiated. The child who understands and prefers to use paper and pencil must listen to, and perform, other subtraction strategies. I guess that is meant to teach tolerance. But how about teaching tolerance by asking all of the children to tolerate starting from the ones?
In MTB, its not okay for the teacher to directly instruct the seated children in how our standard strategy works, but it is okay for children to sit through instruction from their peers. The child who prefers to use counters to count by ones, is left quite alone with his thoughts. How could a teacher intervene, if his strategy is as valid as any other?
A child is not allowed to say, "My way is fast." She is only told that her strategy is "valid." A child is not allowed to find out, "My way is pretty damn slow." He is only told that his strategy is "valid." I hope the girl using paper and pencil doesnt say something like "You are going too slow" to her partner, the boy who is counting by ones in the Difference War game. That might be one of my boys, and the teacher might not notice her impatience, what with all ten pairs of children to oversee.
I do not want a child who resists using a better strategy to be punished or made fun of. His current location on the path to mathematical mastery is a private matter. Discussing it in front of other children is not helpful. What might help is for the teacher to actively teach him. Unfortunately, it must be easier and cheaper for schools to adopt curricula that, for ideological reasons, do not recognize children who would benefit from intervention. We have reading specialists in our schools. Why not math specialists?
Arithmetic is not rote.
The word rote describes the process of memorizing, using repetition without full comprehension, e.g. "to learn by rote".
An arithmetic procedure can be taught by rote, and learned by rote, but the procedure itself is not rote. Any procedure, even those invented by children, can be taught by rote. And it can be taught so badly that children would be better off having never seen it.
Is learning by rote always bad? No. In fact, we learn to count by rote. We learn to count by ones, tens, twos, and fives .
We count collections of objects for our children, e.g. "One school bus, two school bus, three school bus." My toddler understands "one", and "more than one", so that if you hold up three toy school buses, he will say "Two Bus!" His understanding of numbers is not complete. However, I am cheered by what he does understand, and I will continue to teach him to count by rote without worrying about his full comprehension. The words "One, Two, Three" were a gift given to me, and I will give them to my toddler. I will continue to sing "Over in the Meadow" right along with Raffi.
Addition and subtraction are essentially counting up or down "by ones." We can take advantage of our base-ten number system to reduce the effort of counting up or down by ones. The numbers considered in Second Grade are composed from, and therefore can be decomposed into, ones and tens. If a child understands that 16 is 10 + 6, and that 10 + 6 is 16, they are freed to count the tens apart from the ones. Now, there is a choice: start with the ones or start with the tens. Or try it both ways.
If you read no other web article, please read "Basic Skills vs. Conceptual Understanding: A Bogus Dichotomy in Mathematics Education" by H. Wu at the American Federation of Teachers web site, http://www.aft.org/pubs-reports/american_educator/fall99/. While you are there, you can check out Richard Askeys review of Liping Mas book, Knowing and Teaching Elementary Mathematics.
Wu suggests an ideal script for teaching our standard written addition procedure, which could easily be translated for teaching subtraction. There are two main features of his approach. The first is to have children write out the numbers decomposed into tens and ones. Children will be especially ready for this if they have been decomposing numbers to count up or down by tens in their mental arithmetic. The second feature of Wus approach is to give children enough problems to solve on paper that they are receptive to any suggestions that save labor. Children respond to logical consequences.
Wu also suggests scripts for teaching multiplication and the division of fractions. It is very helpful for a parent whose child does not get this same instruction at school.
Another excellent discussion of the value of teaching standard arithmetic procedures is available on the AMS web site, http://www.ams.org/notices/199802/comm-amsarg.pdf.
To summarize, there are two good reasons to do a good job teaching standard arithmetic procedures that "start with the ones:"
For some comic relief, the following selections are from the MTB Teacher Implementation Guide. Do they understand the difference between what is rote and what is arithmetic?
TIG p. 98 "Flexible thinking and mathematical power are our goals, not rote facility with a handful of standard algorithms." Which teacher in what school is teaching our children by rote? Learning math through science experiments with marshmallows in MTB may not lead to all of the flexible thinking and mathematical power that we might have hoped for. First, there is the potential for children to confuse uncertainty in scientific data and interpretation with uncertainty in mathematics. Instead of flexible thinking, you get uncertain thinking. Second, there is the temptation to do only enough mathematics to support the scientific experiment, and no more. That is not true mathematical power. That is called "slamming the door on your way out."
TIG p.177 "The pre-eminence of arithmetic in (an elementary mathematics) curriculum has faded, both because of the availability of calculators and because of the realization that a curriculum focused on rote arithmetic will not meet the needs of students ." Arithmetic develops a childs number sense. Calculators take the place of a childs developing number sense. Calculators do not meet the needs of students in Second Grade.
Proficiency with whole number operations is articulated across grade levels. How a child spends her time in Second Grade affects what she will have time to learn in Fifth Grade. Proficiency with a calculator is not a vertically integrated skill. In Fifth Grade, itll take about five minutes for a child to learn all she needs to know about using a calculator. In Second Grade, our children should put down the calculators. Better to play number games. Or do science experiments with marshmallows. Or read a good book.
TIG p. 178 "The hundreds of hours devoted to arithmetic computation in elementary school leave too little time for other important topics. The traditional approach to rote computation undermines higher-level thinking. Worst of all, the long hours of computational drudgery teach children that mathematics is a most unpleasant business." I agree that teaching computation by rote would be a sad waste of time. Computation is senseless drudgery if our children are never shown why standard procedures work, and why they work so well.
However, I take exception to the assertion that the time spent developing arithmetic proficiency would be better spent on other important topics. Its hard for a child to go on to other important topics if he always has to stop and strategize about the subtraction, or count counters. Proficiency with arithmetic frees your mind for other important thoughts.
If critics of our standard arithmetic procedures would only look more closely, they would see that high-level thinking is employed. It is high-order thinking to understand our so-called low-order arithmetic. But it requires the teacher or the parent to take the time to ask high-order questions on behalf of the children, such as "Why can we start with the ones?" and "What happens if we always start with the tens?"
TIG p. 194 "Pages of problems on the basic facts are not only unnecessary, they can be counterproductive. Students may come to regard mathematics as mostly memorization and may perceive it as meaningless and unconnected to their everyday lives."
My children are a joy to me and to others, but I dont yet trust them to know what is meaningful and connected.
For instance, all 8 year-olds think practicing the piano is meaningless and unconnected to their everyday lives, but that doesnt stop their parents from driving them to piano lessons once a week. Can you imagine a piano teacher who didnt like scales or chords or musical notes printed on a page? A teacher who told you your child was going to invent music?
Many 8 year-olds think broccoli is meaningless and unconnected to their everyday lives, but that doesnt stop parents from serving green vegetables with dinner every night. Have you seen the commercial with the little girl who doesnt want to eat her peas? She is so sweetly sad, her mom cant help but pop open a chocolate nutritional shake, marketed just for kids. My pediatrician told me we were supposed to offer the boys a banana!
Some 8 year-olds think soccer drills are meaningless and unconnected to the game. Why not just play soccer? Truly, if your child has a lot of time and natural talent, that might work. If your child is an average athlete, a few drills might level the playing field for her. Can you imagine a soccer coach whose idea of practice was only to play games? Dont we want coaches who are tough but fair? Coaches that take an interest in each childs ball-handling skills?
I had my boys memorize the Lords Prayer last year. They do not fully comprehend everything in it. But thats okay. They are proud of their ability to recite it. With it they gain entrance into a spiritual community. And they understand enough of the spiritual language so that is has some meaning and connection to their everyday lives. They could not have constructed something so beautiful, nor could I.
Have you been told over and over again that Math Trailblazers is conceptual? I have. The word "conceptual" is an attractive way to describe a learning environment that keeps the arithmetic inside your childs head. To be conceptual is to not get hung up on any one procedure written down on paper with a pencil. Parents should not worry; their child is busy learning mathematical concepts without any distractions printed on pieces of paper.
Advocates of programs like MTB imply that a rigorous approach to written arithmetic in the primary grades is not conceptual. When they say MTB is conceptual, they dont say outright that HMM is not conceptual, they merely imply that it is not. They imply that 8 year-olds cannot develop a complete conceptual understanding of subtraction if we teach them why our standard written procedure for subtraction always works, and why it works so well. It is, after all, printed on a piece of paper.
A concept is a general idea that is derived from and considered apart from what is observed by the senses. The mental process by which this idea is obtained is called abstraction. My toddler is deriving the general idea of "bus" having seen many buses on the road and in picture books and at school and in our toybox. Buses are not always yellow, but they always have many windows. It took him awhile to figure out that taxis are not buses. He still wonders if Maybelle the Cable Car is a bus.
A teacher at our school offered the following example for why she did not like the Houghton-Mifflin approach to teaching mathematics. There is a lesson for First Grade that has all the children in the class look at a number line at the same time, to count up and down on it to add and subtract. The teacher truly believes that number lines are not conceptual. She added that different children might need to use fingers or counters or blocks.
I submit to you that number lines are conceptual. They help a child form the general idea of continuous numbers, that is to say, numbers in between other numbers. You recall the joke about wanting a house with a white picket fence and 2.2 children? There is no such thing as 2.2 children. But there is such a thing as 2.2 candy bars. How about if our children see a number line as an abstraction from candy bars laid end to end?
To help our children move beyond conceiving of numbers as discrete counters, we might show them number lines. A ruler would do the trick. On a ruler, there is a fixed distance between any two adjacent numbers. Counting up and down on a ruler is a geometric way to add and subtract.
Wu asserts in his tutorial, http://www.math.berkeley.edu/~wu/EMI1b.pdf,
"The introduction of the number line is not just a matter of convenience or pedagogy. The number line is one of the most fundamental objects in mathematics. It is, for example, the first step in Descartes introduction of coordinates in space. Its importance cannot be exaggerated."
Have you been told over and over that Math Trailblazers is developmental? I have. The word is used to get parents, the pushy parents who know a little mathematics as well as the worried parents who know a lot of mathematics, to back off. To wait, then wait some more. It becomes a contest to see who can wait the longest. No parent wants to push too hard. But all parents do worry. We want to do enough for our children.
Advocates of programs like MTB imply that a rigorous approach to written arithmetic in the primary grades is not developmental. When they say MTB is developmental, they dont say outright that HMM is not developmental, they merely imply that it is not. They imply that it is not developmentally appropriate to teach 8 year-old children why our standard written procedure for subtraction always works, and why it works so well.
If we understand what is developmentally appropriate practice, we can decide for ourselves if written arithmetic is developmentally inappropriate.
A great place to become acquainted with developmentally appropriate practice is at the National Association for the Education of Young Children (NAEYC) web site, www.naeyc.org. I am a real NAEYC supporter, having enrolled my children in NAEYC-accredited preschools in Oregon and in Colorado that use developmentally appropriate practice and encourage parents to get educated and participate. I was surprised to learn that NAEYC claims to represent childrens interests from birth through age 8, which means they are concerned about Second Grade children, too! Exactly when do kittens turn into cats?
What follows are two selections from the brochure, "Developmentally Appropriate Practice in Early Childhood Programs Serving Children from Birth through 8." Ive simplified the language, emphases are mine, and Ive added editorial comments.
Principles of developmentally appropriate practice:
In summary, NAEYC urges us to move from either/or thinking to both/and thinking, with some examples as follows:
Imagine my surprise when I read further about selecting developmentally appropriate curricula, and NAEYC gave as an example the rejection of any math curriculum that attempts to teach place value concepts to children in the First, Second, or Third grade:
"First, Second, and Third Grade children cannot comprehend place value; they spend hours trying to teach this abstract concept, and children either become frustrated or resort to memorizing meaningless tricks. This is an example of an unrealistic objective that could be attained much more easily later on."
As noted before, one of the excesses of developmentally appropriate practice is that some adults use it to scare other adults into waiting. Wait, then wait some more. Nobody gets to do place value until everybody is ready for place value.
Quite frankly, in an effort to preserve some developmental space for my children, Im more concerned with sheltering them from pernicious messages in our consumer culture to EAT! to BUY! to have SEX! than with sheltering them from the formal study of place value in Second Grade. Somebody get me a WAIVER!
I believe NAEYC borrowed their torrid little speech about place value from Constance Kamii; see Appendix D.
The National Council of Teachers of Mathematics, www.nctm.org, a "big-tent" organization that has published Kamii, came to a different conclusion than Kamii when they put forth their revised Standards in 2000:
"It is absolutely essential that students develop a solid understanding of the base-ten numeration system and place-value concepts by the end of grade 2."
The Reason for the Season of Constructivism in primary mathematics education.
"The teaching of algorithms is based on the erroneous assumption that mathematics is a cultural heritage that must be transmitted to the next generation."
Constance Kamii is a professor of education at the University of Alabama. She studied under Piaget, and she never lets you forget that Piaget taught that logico-mathematical concepts cannot be put into a childs head from outside. Number Lines and Charts and Diagrams, Base-Ten Blocks and Sticks and other Proportionate Manipulatives dont work because they cant work. Children must construct an understanding of number and place value through their own mental arithmetic activity. Kamii defines constructivism at http://www.terc.edu/investigations/relevant/html/constructivistlearning.html. For a critical review of constructivist teaching and situated learning, see John Andersons paper at http://act.psy.cmu.edu/personal/ja/misapplied.html.
Although I have not purchased Kamiis books on how children reinvent arithmetic, I have read her paper, "Teaching Place Value and Double-Column Addition" available at http://www.enc.org/professional/learn/research/journal/math/document.shtm?input=ACQ-104912-4912_48#3. Some interesting features of Kamiis teaching emerge:
The only arithmetic allowed in her classroom is mental arithmetic.
In a Kamii classroom, the children read and solve problems that the teacher writes in correct mathematical sentences on the chalkboard. But the children do not write anything down on paper with a pencil. Textbooks and workbooks are absent from her classroom. Even manipulatives are off-limits, such as the base-ten blocks that are popular in MTB. Kamii teaches that these manipulatives are external to the child, and cannot be relied upon as a means for the child to internalize concepts of number and place value. The child must solve the problems in her head and give the answers verbally.
Increasing a childs capacity for mental arithmetic with numbers up to 100 is laudable.
They think it, they speak it. The teacher writes it. It reminds me of language immersion. But it does come at the expense of having children start learning to write mathematical sentences by themselves that can be understood by anyone.
Imagine that we didnt teach writing in K-2. No "writing road to reading." Not even invented spellings. The children only played games with letters and words in Kindergarten and First Grade, and then played games and had class discussions about what the teacher writes on the chalkboard in Second Grade.
Kamiis emphasis on mental math alone has enabled me to see most clearly the difference between MTB and HMM. I was trying to compare the writing of mathematics between MTB and HMM, and I couldnt figure out why MTB set great store by journaling in bad, but authentic, English. Now I understand that they were trying to get away from having the children write correct mathematical sentences because they want to keep the emphasis on mental arithmetic. Of course, MTB employs manipulatives and 200 charts and calculators. But now that I have read Kamiis conception of the K-2 classroom I can understand what MTB attempts to do and why they fail to do it.
Mental arithmetic never starts with the ones.
Whats the advantage of Left to Right Arithmetic? Certainly, when were looking at numbers between one and nine, there is no left or right. Its when we go past nine that there is suddenly a left and a right. The left is the "tens" and the right is the "ones".
Kamii gives some good examples of how children can perform their mental addition. She notes that all of the children in her classroom start with the tens.
In mental arithmetic, often easier for us to add starting from the left, as long as the number remains small. Think of yourself at the shoe store. Youve just picked out a pair of shoes that cost $36. Youd like to take advantage of a sale and get the second pair of the same shoes for half price, but you left your checkbook and credit cards at home. You have three 20s and two 1s in your purse, and you promised youd stop for fast food on the way home. It takes about $10 to feed your family. You do the arithmetic in your head, and you start with the tens because you have nothing to write with. If it was me, I would calculate thus: 36 + 18 = 46 + 8 = 50 + 4 = 54. And 62 54 = 2 + 6 = 8. Not enough money for fast food.
The subject of mental strategies has been studied extensively, and there is a lot of research published on it. I used an "N10" strategy to find the cost of two pairs of shoes. In the Netherlands, they have taught Second Grade children to perform addition and subtraction starting with the tens for many years. They use textbooks and manipulatives and number lines. Amazingly enough, the children still make a lot of arithmetic mistakes. More research is then conducted to try to figure out what new manipulative or number line could be employed to help children make fewer mistakes starting with the tens.
Teachers must ask the right questions at the right time, and teachers must avoid saying that an answer is right or wrong.
Although Kamii claims that children can construct their own logico-mathematical relationships, teachers are essential to ask children to consider the right arithmetic problem at the right time. Without such teachers, our children would have few opportunities to reinvent arithmetic.
Teachers are also encouraged to disagree or agree with a childs explanation of their mental arithmetic, but they should do it politely and to explain themselves:
"The teacher is careful not to be the omniscient authority who decrees what is right or wrong. She or he only agrees or disagrees and tries to present another point of view on an equal footing."
"The teacher calls on individual children and writes all the answers that are given by them. Being careful not to say that an answer is right or wrong, the teacher then asks for an explanation of each answer or procedure invented by the children."
I think that is the best way to correct children let them do it themselves. However, the teacher needs to commit to this corrective procedure on behalf of all twenty children in the classroom. The Socratic method requires teachers to continue asking questions until the truth comes out. The AMS, in their discussion with NCTM on the value of mathematical algorithms, kindly refers to this as "mathematical closure."
Teachers must speak correctly and demand the same of the children.
The premise of Kamiis academic career is that children do not understand place value when they are taught by "traditional," "rote" methods that try to "put" place value in a childs head. To prove that children do not understand place value, she interviews them and asks them about the number 16. You can read the interview details at www.enc.org.
Her interview is unforgiving; a child is supposed to offer the units without prompting, and the child fails to convey any understanding of place value if the child fails to offer the units. It is well worth your time to conduct her interview with your own child, and also to extend the interview to get more complete information about your childs understanding of place value.
Kamii claims that no Kindergarteners from "traditional" classrooms say that the 1 means ten. Kamii claims that a Third of Second Graders and Half of Fourth Graders say that the 1 means ten. As any pollster can tell you, you get the result you want by carefully choosing the words in the question.
When the interviewer circles the number 1 in the number 16, isolating it, and asks the child "what does this part mean? Can you show me with your chips?" I assert that a child is absolutely correct to say "one". "One" is the coefficient of the tens. Let me put it this way. If I write down the name, "Becky Doe" and I circle "Becky" and I ask you "what does this part mean?" and you say it means "Becky," you are absolutely right. If I ask you "what does this part of the name mean?" you might even think to say, "It means her first name is Becky".
Im extremely disappointed that Kamii never asks children "what does this part of the number mean?"
Id like to borrow an example that Kamii gives from her classroom. The children are considering the problem 36 + 46. A child says "three and four are seventy." The child does not nor ever will believe that 3 + 4 = 70. The child merely left out the units, the word "tens." It is entirely natural to omit the units when you are sure of what you are doing that you can treat addition of tens as single digit addition. Of course, its best for the teacher to remind the child to say "three tens" and "four tens" or "thirty and forty" for the benefit of every child in the classroom.
In a Kamii classroom, the teacher doesnt let a child get sloppy and say "three and four are seventy." The teacher always asks the child to keep track of the units.
On the first Monday of Spring Break, I sat down with Tom and Luke at the kitchen table. I poured out a pile of Cheerios and asked Tom to not say anything. I wrote 27 on a piece of paper and asked Luke to count out that many Cheerios. He did. I asked him "what does this part mean?" and he said 7. I asked him to count out that many. Then I asked him "what does this part mean?" and he said 2. Tom immediately interjected "2 Tens!" and Luke corrected himself and said "2 Tens." I scolded Tom. In educator-speak, Tom "elicited the multidigit context" for Luke.
Then I set about to see where Toms knowledge of place value would fail him. I wrote down 14 and 23 columnwise as to add them. I asked Tom to count out this many and that many, and he did. I asked him what this part meant and what that part meant. He got it all right. So I tried to get a little tricky. I asked him, what do I get if I put this part (the 1 from 14) and that part (the 3 from 23) together. He said "1 and 3 is four" and then he immediately corrected himself and said "1 ten and 3 is ten, twenty, thirty, forty!" and I said "think about it." He then said "1 ten and 3 is thirty!" and I asked him to think about it some more and he said "thirteen!" In educator-speak, that means he had 13 "available as his fourth meaning." Then I asked Tom about the other cross parts (the 2 from 23 and the 4 from 14) and he correctly said "2 tens and 4 is twenty-four!"
I know the sample size is extremely small, but fully Half of the Kindergarteners at my house understand place value. My boys have had no special mathematics instruction. They celebrated the 100th day of Kindergarten; their teacher taught them to count by tens to 100. We do a little bit of single-digit cookie math at the kitchen table.
Karen Fuson, a researcher from Northwestern University, suggests the following extension to Kamiis interview. If a child shows one chip at first, ask "what else could this part mean?" or ask the child to "look at the places again." This extension is detailed in Fusons 1990 article regarding the use of base-ten blocks to teach place value concepts. Fuson found that before extending the interview question, 12 out of 22 children showed ten chips. By extending the interview question for children who showed one chip, 8 more children showed ten chips.
Parting thoughts about Constance Kamii
In her paper "The Harmful Effects of Algorithms in Grades 1 4," Kamii ends with this thought:
"Teaching algorithms to (struggling) students sent them the message that the logic of this procedure is too much for you; so just follow these steps and youll get the right answer."
Constance Kamii and constructivist curricula such as Kendall/Hunts Math Trailblazers have sent a clear message to teachers that the logic of our standard written procedures is too much for Second Grade children. Dont even try to teach it. All a teacher can do is put a worked subtraction problem on the overhead and see if it causes any child to construct a private logico-mathematical understanding of "starting with the ones."
I wish Kamii would apply her formidable talents to teaching teachers how to speak correctly and ask the right questions, to instruct children in the logic of our standard written procedures. Her insistence on construction blinds her to the possibilities of instruction.
Must teachers reinvent instruction, or does Kamii think they can be directly taught how to teach? What happens if a teacher learns a constructivist approach, such as that in MTB, "by rote?" How many teachers have taught from constructivist curricula without ever asking high-order questions, such as "what happens if we dont tell children about starting with the ones?" and "what if we never start with the ones?"
If you read more of Kamiis teachings, as well as educational research that compares good new methods with the bad old methods, you should be aware of a phenomenon called the Hawthorne effect: an increase in worker productivity produced by the psychological stimulus of being singled out and made to feel important.
I ran across a reference to the Hawthorne effect in a paper by Hiebert and Wearne. They were studying the effects of teachers asking high-order questions of children learning addition and subtraction in the Second Grade. The improved academic results obtained by the "high-order teachers" were compared with results obtained by teachers left to their own devices. Hiebert and Wearne point out that:
"Working with specially employed teachers permitted a faithful implementation of an alternative instructional approach but confounded teachers with treatment. This means that the Hawthorne effect cannot be ruled out."